IMDAD ULLAH KHAN UMM AL-QURA UNIVERSITY June 12, 2013 Hypergraphs - PowerPoint PPT Presentation

PERFECT MATCHINGS IN UNIFORM HYPERGRAPHS IMDAD ULLAH KHAN UMM AL-QURA UNIVERSITY June 12, 2013

Hypergraphs A hypergraph H is a family of subsets ( E ( H )) of a ground set V ( H ) H = ( V , E ) | V ( H ) | = n 1 2 3 6 H := E 5 4 V = { 1 , 2 , 3 , 4 , 5 , 6 } E = {{ 1 , 5 } , { 1 , 2 , 3 } , { 2 , 4 , 5 } , { 1 , 4 , 5 , 6 }}

Hypergraphs: Terminology A hypergraph is k -uniform if all edges are k -sets � V � H = ( V , E ), E ⊆ k 1 k -graphs 2 3 6 2-graphs are graphs 5 4 A k -graph is complete if all k -sets are edges � V � H = ( V , E ), E = k

Hypergraphs: Terminology H ( V 1 , V 2 , . . . , V k ) is a k -partite k -graph, if V 1 , V 2 , . . . , V k is a partition of V ( H ) V 1 Each edge uses one vertex from each part V 2 V 3 Complete k -partite k -graph Balanced complete k -partite k -graph, K r ( t ), t : size of each part.

Hypergraphs: Matching A matching in a hypergraph is a set of disjoint edges A perfect matching is a matching that covers all the vertices � n � edges in k -graphs k n ∈ k Z

Hypergraphs: Degrees H : k -graph, 1 ≤ d ≤ k − 1 � V � S ∈ d Degree of S is the number of edges containing S d H ( S ) = |{ e ∈ E : S ⊂ e }| minimum d -degree, δ d ( H ) = min d H ( S ) S ∈ ( V d ) d = k − 1: δ k − 1 ( H ) minimum co-degree δ 2 ( H ) = 1 d = 1: δ 1 ( H ) minimum vertex degree δ 1 ( H ) = 2

Degree Threshold for Perfect Matching Sufficient conditions to ensure existence of perfect matching Definition m d ( k , n ) = min { m : δ d ( H ) ≥ m = ⇒ H has a PM } Theorem m 1 (2 , n ) ≤ n 2 n 2 − 1 K n/ 2+1 ,n/ 2 − 1 n 2 + 1 Result is best possible: m 1 (2 , n ) = n 2 .

Perfect Matching: codegree A | A | odd even | A | ∼ n 2 δ 3 ( H ) ∼ n 2 − k B

Perfect Matching: codegree Theorem 1 K¨ uhn-Osthus 2006 2 + 3 k 2 √ n log n m k − 1 ( k , n ) ≤ n 2 R¨ odl-Ruci´ nski-Szemer´ edi 2006 A m k − 1 ( k , n ) ≤ n | A | odd 2 + C log n even | A | ∼ n 2 3 R¨ odl-Ruci´ nski-Schacht-Szemer´ edi 2008 δ 3 ( H ) ∼ n 2 − k B m k − 1 ( k , n ) ≤ n 2 + k / 4 4 R¨ odl-Ruci´ nski-Szemer´ edi 2009 m k − 1 ( k , n ) ≥ n 2 − k + { 3 2 , 2 , 5 2 , 3 }

Perfect Matching: d -degree Theorem (Pikhurko 2008) For k 2 ≤ d ≤ k − 1 � 1 � � n − d � m d ( k , n ) ≤ 2 + ǫ k − d Theorem (Treglown-Zhao 2012) For k 2 ≤ d ≤ k − 1 � n − d � m d ( k , n ) ∼ 1 2 k − d

Perfect Matching: vertex-degree | A | = n 3 − 1 A B � n − 1 � 2 n/ 3 � � δ 1 ( H ) = − 2 2 Conjecture � n − n � n − d � k + 1 − d � 1 ≤ d < k / 2 m d ( k , n ) ∼ − k − d k − d � � k − d � � n − d � k − 1 � 1 ≤ d < k / 2 m d ( k , n ) ∼ 1 − k − d k

Perfect Matching: Vertex Degree Conjecture � � k − d � � n − d � k − 1 � 1 ≤ d < k / 2 m d ( k , n ) ∼ 1 − k k − d Theorem (H` an-Person-Schacht 2009) d < k � k − d � � n − d � m d ( k , n ) ≤ + ǫ 2 k − d k Theorem (Markstr¨ om-Ruci´ nski 2010) � k − d 1 � � n − d � d < k m d ( k , n ) ≤ − k k − 1 + ǫ 2 k k − d

Perfect Matching: Vertex Degree Conjecture � � k − d � � n − d � k − 1 � 1 ≤ d < k / 2 m d ( k , n ) ∼ 1 − k k − d k = 3 , d = 1 → 5 9 . k = 4 , d = 1 → 37 64 . k = 5 , d = 1 → 369 625 . Theorem (H` an-Person-Schacht 2009) � 5 � � n � m 1 (3 , n ) ≤ 9 + ǫ 2 Theorem (Markstr¨ om- Ruci´ nski 2010) � 42 � � n � m 1 (4 , n ) ≤ 64 + ǫ 3

Perfect Matching: Vertex Degree Theorem (K.) If H is a 3 -graph on n ≥ n 0 vertices and � n − 1 � � 2 n / 3 � δ 1 ( H ) ≥ − + 1 , 2 2 then H contains a perfect matching. | A | = n 3 − 1 A B � n − 1 � 2 n/ 3 � � δ 1 ( H ) = − 2 2 Independently, K¨ uhn-Osthus-Treglown proved this. In fact they proved a stronger result.

Perfect Matching: Vertex Degree Theorem (K¨ uhn-Osthus-Treglown) If H is a 3 -graph on n ≥ n 0 vertices, 1 ≤ m ≤ n / 3 , and � n − 1 � � n − m � δ 1 ( H ) ≥ − + 1 , 2 2 then H contains a matching of size at least m. | A | = m − 1 A B � n − 1 � n − m � � δ 1 ( H ) = − 2 2

Perfect Matching: Vertex Degree Theorem (K.) If H is a 4 -graph on n ≥ n 0 vertices and � n − 1 � � 3 n / 4 � δ 1 ( H ) ≥ − + 1 , 3 3 then H contains a perfect matching. | A | = n 4 − 1 A B � n − 1 � 3 n/ 4 � � δ 1 ( H ) = − 3 3

Perfect Matching: Vertex Degree Theorem (Alon-Frankl-Huang-R¨ odl-Ruci´ nski-Sudakov 2012) � n − 1 m 1 (4 , n ) ∼ 37 � 64 3 � n − 2 m 2 (5 , n ) ∼ 1 � 2 3 � n − 1 m 1 (5 , n ) ∼ 369 � 625 4 � n − 2 671 � m 2 (6 , n ) ∼ 1296 4 � n − 3 m 3 (7 , n ) ∼ 1 � n 3

Perfect Matching: Vertex Degree Theorem (K.) If H is a 3 -graph on n ≥ n 0 vertices and � n − 1 � � 2 n / 3 � δ 1 ( H ) ≥ − + 1 , 2 2 then H contains a perfect matching. | A | = n 3 − 1 A B � n − 1 � 2 n/ 3 � � δ 1 ( H ) = − 2 2

3 -graphs - vertex degree: Proof Strategy We consider two cases 1 H is close to the extremal construction 2 H is non-extremal | A | ∼ n 3 − 1 A very few such edges are in H B

3 -graphs - vertex degree: Absorbing Absorbing Technique S ⊂ V , A matching M absorbs the set S if ∃ M ′ : V ( M ′ ) = V ( M ) ∪ S . s 1 M s 2 S M ′ s 3

3 -graphs - vertex degree: Absorbing Lemma Absorbing Lemma (H` an-Person-Schacht 2009) � 1 � � n � If δ 1 ( H ) ≥ 2 + ǫ , then k ∃ M A such that | V ( M A ) | = ǫ 1 n and ∀ S : | S | = ǫ 2 n , M A is S -absorbing.

3 -graphs - vertex degree: Proof Overview Proof Outline: 1 Find a small absorbing matching M A ( | V ( M A ) | ≤ ǫ 1 n ) 2 Find an almost perfect matching M ′ in H − V ( M A ) V 0 = V ( H ) − ( V ( M A ) + V ( M ′ ) | V 0 | ≤ ǫ 2 n 3 Absorb V 0 into M A M A M ′ V 0

3 -graphs - vertex degree: Almost Perfect Matching: M A M ′ V 0 We cover almost all graph with complete tripartite graphs Theorem (Erd˝ os 1964) , then H has K 3 ( c √ log n ) . � n � If | E ( H ) | ≥ ǫ 3 Using this find as many K 3 ( t )’s. I . . . very few edges Extend this to almost perfect cover.

3 -graphs - vertex degree: Almost perfect cover I . . . very few edges Extend this to almost perfect cover. Suppose many pairs in I make edges with many vertices in two color classes of many tripartite graphs. I I

3 -graphs - vertex degree: Almost perfect cover I . . . very few edges Extend this to almost perfect cover. The link graph: I

3 -graphs - vertex degree: Almost perfect cover Fact Let B be a balanced bipartite graph on 6 vertices. If B has at least 5 edges, then B has a perfect matching or B contains B 320 as a subgraph or B is isomorphic to B 311 . B 320 B 311 PM

3 -graphs - vertex degree: Almost perfect cover I . . . very few edges Few edges inside I and few pairs in I make edges with vertices in two color classes of many tripartite graphs. δ 1 ( H ) implies that on average the link graph of a pair of tripartite graphs has 5 edges. Suppose for many pairs the link graph has perfect matching. T i I T j

3 -graphs - vertex degree: Almost perfect cover I . . . very few edges Suppose for many pairs the link graph has a B 320 . T i I T j

3 -graphs - vertex degree: Almost perfect cover I . . . very few edges Few edges inside I . Few pairs in I make edges with vertices in two color classes of many tripartite graphs. δ 1 ( H ) implies that on average the link graph of a pair of tripartite graphs has 5 edges. For few pairs the link graph has perfect matching or has a B 320 . V 1 . . . . . . V 2 V 3

3 -graphs - vertex degree: Almost perfect cover I . . . very few edges For almost all pairs of tripartite graphs, the link graph is ismorphic to B 311 . V 1 . . . V 2 I V 3 Few edges in V 2 ∪ V 3 Similarly few edges with two vertices in I and one in V 2 ∪ V 3 .

3 -graphs - vertex degree: Almost perfect cover V 1 . . . V 2 I V 3 Few edges in I Few edges in V 2 ∪ V 3 Few edges with two vertices in I and one in V 2 ∪ V 3 . By definition of B 311 , few edges with one vertex in I and two in V 2 ∪ V 3 . So few edges in V 2 ∪ V 3 ∪ I , while its size is ∼ 2 n / 3, hence H is close to the extremal construction.

3 -graphs - vertex degree: Almost perfect cover Thank You!